Advertisement

Turing Bifurcations and Pattern Selection

  • P. Borckmans
  • G. Dewel
  • A. De Wit
  • D. Walgraef
Part of the Understanding Chemical Reactivity book series (UCRE, volume 10)

Abstract

Pattern forming instabilities in spatially extended dissipative systems driven away from equilibrium have been the focus of a large activity for many years. The goal of this chapter is to present some theoretical concepts that have been developed to understand and describe these dissipative structures [1] from a macroscopic point of view. Although these methods present generic features we shall only be concerned with chemical patterning and shall not discuss here instabilities in hydrodynamics, liquid crystals and nonlinear optics that all present similar types of organization because the latter have been the subject of recent well-documented reviews [2–5]. Moreover, we essentially consider the self-organization of structures discarding the spatial patterning resulting from boundary conditions.

Keywords

Bifurcation Diagram Linear Stability Analysis Bifurcation Parameter Pattern Selection Amplitude Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Nicolis, G. and Prigogine, I., Self-Organization in Nonequilibrium Systems ( Wiley, New York, 1977 ).Google Scholar
  2. 2.
    Manneville, P., Dissipative Structures and Weak Turbulence ( Academic Press, San Diego, 1990 ).Google Scholar
  3. 3.
    Kai, S., Physics of Pattern Formation in Complex Dissipative Systems ( World Scientific, Singapore, 1992 ).Google Scholar
  4. 4.
    Cross, M. C. and Hohenberg, P. C., Rev. Mod. Phys. 65, 854 (1993).CrossRefGoogle Scholar
  5. 5.
    Newell, A. C. and Moloney, J. V., Nonlinear Optics ( Addison-Wesley, Redwood City, 1992 ).Google Scholar
  6. 6.
    Zeldovich, Y. B., Theory of Combustion and Detonation of Gases ( Academy of Sciences USSR, Moscow, 1944 ).Google Scholar
  7. 7.
    Rashevski, N., Mathematical Biophysics ( University of Chicago Press, Chicago, 1938 ).Google Scholar
  8. 8.
    Turing, A., Philos. Trans. R. Soc. Lond. B 237, 37 (1952).CrossRefGoogle Scholar
  9. 9.
    Glansdorff, P. and Prigogine, I., Thermodynamic Theory of Structure, Stability and Fluctuations ( Wiley, New York, 1971 ).Google Scholar
  10. 10.
    Meinhardt, H., Models of Biological Pattern Formation ( Academic Press, New York, 1982 ).Google Scholar
  11. 11.
    Murray, J. D., Mathematical Biology ( Springer, Berlin, 1989 ).Google Scholar
  12. 12.
    Kerner, B. S. and Osipov, V. V., Soy. Phys. JETP 47, 874 (1978).Google Scholar
  13. 13.
    Willebrand, H., Niedernostheide, F. J., Dohmen, R., and Purwins, H. G., in Oscillations and Morphogenesis, edited by L. Rensing ( Dekker, New York, 1992 ), p. 81.Google Scholar
  14. 14.
    Niedernostheide, F. J., Dohmen, R., Willebrand, H., Schulze, H. J., and Purwins, H. G., in Nonlinearity with Disorder, edited by K. Abdullaev, A. R. Bishop, and S. Pneumatikos ( Springer-Verlag, Berlin, 1992 ), p. 282.CrossRefGoogle Scholar
  15. 15.
    Falta, J., Imbihl, R., and Henzler, M., Phys. Rev. Lett. 64, 1409 (1990).CrossRefGoogle Scholar
  16. 16.
    Krishan, K., Nature 287, 420 (1980).CrossRefGoogle Scholar
  17. 17.
    Walgraef, D. and Ghoniem, N. M., Phys. Rev. B 13, 8867 (1989).CrossRefGoogle Scholar
  18. 18.
    Emelyanov, V. I., Laser Physics 2, 389 (1992).Google Scholar
  19. 19.
    Castets, V., Dulos, E., Boissonade, J., and De Kepper, P., Phys. Rev. Lett. 64, 2953 (1990).CrossRefGoogle Scholar
  20. 20.
    Ouyang, Q. and Swinney, H. L., Nature 352, 610 (1991).CrossRefGoogle Scholar
  21. 21.
    Kerner, B. S. and Osipov, V. V., in Nonlinear Waves I, edited by A. V. Gaponov-Grekov, M. I. Rabinovich, and J. Engelbrecht ( Springer-Verlag, Berlin, 1989 ), p. 126.CrossRefGoogle Scholar
  22. 22.
    Elmer, F. J., Physica D 30, 321 (1988);CrossRefGoogle Scholar
  23. Elmer, F. J., Phys. Rev. A 41, 4174 (1990).CrossRefGoogle Scholar
  24. 23.
    Middya, U., Sheintuch, M., Graham, M. D. and Luss, D., Physica D 63, 393 (1993).CrossRefGoogle Scholar
  25. 24.
    Bedeaux, D., Mazur, P., and Pasmanter, R. A., PhysicaA 86, 355 (1977).CrossRefGoogle Scholar
  26. 25.
    Ertl, G., Science 254, 1750 (1991).CrossRefGoogle Scholar
  27. 26.
    Dewel, G., Borckmans, P., and Walgraef, D., in Chemical Instabilities, edited by G. Nicolis and F. Baras ( Reidel, Dordrecht, 1984 ), p. 385.CrossRefGoogle Scholar
  28. 27.
    Kerner, B. S. and Osipov, V. V., Sov. Phys. Usp. 33, 679 (1991).CrossRefGoogle Scholar
  29. 28.
    Newell, A. C., Passot, T., and Lega, J., Annu. Rev. Fluid Mech. 25, 399 (1993).CrossRefGoogle Scholar
  30. 29.
    Pomeau, Y., Physica D 23, 3 (1986).CrossRefGoogle Scholar
  31. 30.
    Ortoleva, P. and Ross, J., J. Chem. Phys 63, 3398 (1975).CrossRefGoogle Scholar
  32. 31.
    Thual, O. and Fauve, S., J. Phys. (Paris) 49, 1829 (1988).CrossRefGoogle Scholar
  33. 32.
    Busse, F., Rep. Progr. Phys. 41, 1929 (1978).CrossRefGoogle Scholar
  34. 33.
    Malomed, B. A. and Tribelskii, M. I., Soy. Phys. JETP 65, 305 (1987).Google Scholar
  35. 34.
    Epstein, I. R., Lengyel, I., Kadar, S., Kagan, M., and Yokoyama, M., Physica A 188, 26 (1992).CrossRefGoogle Scholar
  36. 35.
    Haken, H. and Olbrich, H., J. Math. Biol. 6, 317 (1978).CrossRefGoogle Scholar
  37. 36.
    Walgraef, D., Dewel, G., and Borckmans, P., Phys. Rev. A 21, 397 (1980).CrossRefGoogle Scholar
  38. 37.
    Pismen, L. M., J. Chem. Phys. 72, 1900 (1980).CrossRefGoogle Scholar
  39. 38.
    Walgraef, D., Dewel, G., and Borckmans, P., Adv. Chem. Phys. 49, 311 (1982).CrossRefGoogle Scholar
  40. 39.
    Metens, S., Dewel, G., and Borckmans, P., ‘Pattern Selection in a 2D Reaction-Diffusion System’, preprint (1993).Google Scholar
  41. 40.
    Borckmans, P., De Wit, A., and Dewel, G., Physica A 188, 137 (1992).CrossRefGoogle Scholar
  42. 41.
    Dufiet, V. and Boissonade, J., J. Chem. Phys. 96, 664 (1992).CrossRefGoogle Scholar
  43. 42.
    Dufiet, V. and Boissonade, J., Physica A 188, 158 (1992).CrossRefGoogle Scholar
  44. 43.
    Jensen, O., Pannbacker, V. O., Dewel, G., and Borckmans, P., Phys. Lett. A 179, 91 (1993).CrossRefGoogle Scholar
  45. 44.
    Pearson, J. E., Science 261, 189 (1993).CrossRefGoogle Scholar
  46. 45.
    Lengyel, I. and Epstein, I. R., Science 251, 650 (1991).CrossRefGoogle Scholar
  47. 46.
    Verdasca, J., De Wit, A., Dewel, G., and Borckmans, P., Phys. Lett. A 168, 194 (1992).CrossRefGoogle Scholar
  48. 47.
    Ouyang, Q., Communication at the 76th Canadian Society for Chemistry Conference, Sherbrooke (1993).Google Scholar
  49. 48.
    Walgraef, D., Dewel, G., and Borckmans, P., Nature 218, 606 (1985).CrossRefGoogle Scholar
  50. 49.
    Malomed, B. A., Nepomnyashchii, A. A., and Tribelskii, M. I., Soy. Phys. JETP 69, 388 (1990).Google Scholar
  51. 50.
    Newell, A. C. and Pomeau, Y., J. Phys. A: Math. Gen. 26, L429 (1993).CrossRefGoogle Scholar
  52. 51.
    Perraud, J. J., Agladze, K., Dulos, E., and De Kepper, P., Physica A 188, 1 (1992).CrossRefGoogle Scholar
  53. 52.
    De Wit, A., Dewel, G., Borckmans, P., and Walgraef, D., Physica D 61, 289 (1993).CrossRefGoogle Scholar
  54. 53.
    Pomeau, Y. and Manneville, P., J. Phys. Lettres (Paris) 40, 609 (1979).CrossRefGoogle Scholar
  55. 54.
    Sakaguchi, H., Progr. Theor. Phys. 89, 1123 (1993).CrossRefGoogle Scholar
  56. 55.
    Hoyle, R. B., Physica D 67, 198 (1993).CrossRefGoogle Scholar
  57. 56.
    Malomed, B. A., Nepomnyashchii, A. A., and Nuz, A. E., Physica D 70, 357 (1994).CrossRefGoogle Scholar
  58. 57.
    Gunaratne, G. H., Phys. Rev. Lett. 71, 1367 (1993).CrossRefGoogle Scholar
  59. 58.
    Lauzeral, J., Metens, S., and Walgraef, D., Europhys. Lett. 24, 707 (1993).CrossRefGoogle Scholar
  60. 59.
    Sushchik, M. M. and Tsimring, L. S., Physica D 74, 301 (1994).CrossRefGoogle Scholar
  61. 60.
    Landau, L. D. and Lifshitz, E. M., Theory of Elasticity ( Pergamon, Oxford, 1970 ).Google Scholar
  62. 61.
    Cross, M. C. and Newell, A. C., Physica D 10, 299 (1984).CrossRefGoogle Scholar
  63. 62.
    Passot, T. and Newell, A. C., Physica D 74, 301 (1994).CrossRefGoogle Scholar
  64. 63.
    Auchmuty, J. F. and Nicolis, G., Bull. Math. Biol. 37, 323 (1975).Google Scholar
  65. 64.
    Herschkowitz-Kaufman, M. and Nicolis, G., J. Chem. Phys. 56, 1890 (1972).CrossRefGoogle Scholar
  66. 65.
    Dewel, G. and Borckmans, P., Phys. Lett. A 138, 189 (1989).CrossRefGoogle Scholar
  67. 66.
    Walton, I. C., Quart. J. Mech. (Appl. Math.) 35, 33 (1982).CrossRefGoogle Scholar
  68. 67.
    Dewel, G., Walgraef, D., and Borckmans, P., J. Chimie Physique 84, 1335 (1987).Google Scholar
  69. 68.
    Kramer, L., Ben-Jacob, E., Brand, H., and Cross, M. C., Phys. Rev. Lett. 49, 1891 (1982).CrossRefGoogle Scholar
  70. 69.
    Ahlers, G., Physica D 51, 421 (1991).CrossRefGoogle Scholar
  71. 70.
    Walton, I. C., Stud. Appl. Math. 67, 199 (1982).Google Scholar
  72. 71.
    Malomed, B. A. and Nepomnyashchii, A. A., Europhys. Lett. 21, 195 (1993).CrossRefGoogle Scholar
  73. 72.
    Walton, I. C., J. Fluid Mech. 131, 455 (1983).CrossRefGoogle Scholar
  74. 73.
    De Wit, A., Borckmans, P., and Dewel, G., in Instabilities and Nonequilibrium Structures IV, edited by E. Tirapegui and W. Zeller ( Kluwer, Dordrecht, 1993 ), p. 247.CrossRefGoogle Scholar
  75. 74.
    Ortoleva, P. and Ross, J., J. Chem. Phys. 56, 4397 (1972).CrossRefGoogle Scholar
  76. 75.
    Bimpong-Bota, E. K., Nitzan, A., Ortoleva, P., and Ross, J., J. Chem. Phys. 66, 3650 (1977).CrossRefGoogle Scholar
  77. 76.
    Bensimon, D., Shraiman, B. I. and Croquette, V., Phys. Rev. A 38, 5461 (1988).CrossRefGoogle Scholar
  78. 77.
    Malomed, B. A., Nepomnyashchii, A. A., and Tribelskii, M. I., Phys. Rev. A 42, 7244 (1990).CrossRefGoogle Scholar
  79. 78.
    Kolodner, P., Phys. Rev. E 48, R4187 (1993).CrossRefGoogle Scholar
  80. 79.
    Peyrard, M. and Kruskal, M. D., Physica D 14, 88 (1984), and references therein.Google Scholar
  81. 80.
    Coullet, P. and Emilson, K., Physica A 188, 190 (1992).CrossRefGoogle Scholar
  82. 81.
    Gaponov-Grekov, A. V., Lomov, A. S., Osipov, G., and Rabinovich, M. I., in Nonlinear Waves I, edited by A. V. Gaponov-Grekov, M. I. Rabinovich, and J. Engelbrecht ( Springer-Verlag, Berlin, 1989 ), p. 65.CrossRefGoogle Scholar
  83. 82.
    Jones, C. A. and Proctor, M. R., Phys. Lett. A 121, 224 (1987).CrossRefGoogle Scholar
  84. 83.
    Marques, C. M. and Cates, M. E., Europhys. Lett. 13, 267 (1990).CrossRefGoogle Scholar
  85. 84.
    Leibler, L., Macromolecules 13, 1602 (1980).CrossRefGoogle Scholar
  86. 85.
    Thomas, E. L., Anderson, D. M., Henkee, C. S., and Hoffman, D., Nature 334, 598 (1988).CrossRefGoogle Scholar
  87. 86.
    Anderson, D. M., Davis, H. T., Scriven, L. E. and Nitsche, J. C., Adv. Chem. Phys. 77, 337 (1990).CrossRefGoogle Scholar
  88. 87.
    Charvolin, J. and Sadoc, J. F., J. Physique (Paris) 48, 1559 (1987).CrossRefGoogle Scholar
  89. 88.
    Andersson, S., Hyde, S. T., Larsson, K., and Lidin, S., Chem. Rev. 88, 221 (1988).CrossRefGoogle Scholar
  90. 89.
    Olvera de la Cruz, M., Phys. Rev. Lett. 67, 85 (1991).CrossRefGoogle Scholar
  91. 90.
    De Wit, A., unpublished results.Google Scholar
  92. 91.
    Perraud, J. J., De Wit, A., Dulos, E., De Kepper, P., Dewel, G., and Borckmans, P., Phys. Rev. Lett. 71, 1272 (1993).CrossRefGoogle Scholar
  93. 92.
    Hunding, A. and Sorensen, P. G., J. Math. Biol. 26, 27 (1988).CrossRefGoogle Scholar
  94. 93.
    Lengyel, I. and Epstein, I. R., Proc. Natl. Acad. Sci. USA 89, 3977 (1992).CrossRefGoogle Scholar
  95. 94.
    Kidachi, H., Progr. Theor. Phys. 63, 1152 (1980).CrossRefGoogle Scholar
  96. 95.
    De Wit, A., Ph.D. Thesis, Université Libre de Bruxelles (1993).Google Scholar
  97. 96.
    Keener, J. P., Stud. Appl. Math. 55, 187 (1976).Google Scholar
  98. 97.
    Rovinsky, A. B. and Menzinger, M., Phys. Rev. A 46, 6315 (1992).CrossRefGoogle Scholar
  99. 98.
    Borckmans, P., Jensen, O., Pannbacker, V. O., Mosekilde, E., Dewel, G., and De Wit, A., in Dynamical Phenomena in Living Systems,edited by E. Mosekilde and O. Mouritsen (Synergetics-Springer-Verlag, Berlin) (to appear).Google Scholar
  100. 99.
    Pannbacker, V. O., Jensen, O., Mosekilde, E., Dewel, G., and Borckmans, P., in SpatioTemporal Patterns in Nonequilibrium Complex Systems,edited by P. Palffy-Muhoray and P. Cladis (to appear).Google Scholar
  101. 100.
    De Kepper, P., Perraud, J. J., Rudovics, B., and Dulos, E., Int. J. of Bifurcation and Chaos (to appear).Google Scholar
  102. 101.
    Kolodner, P., Phys. Rev. E 48, R665 (1993).CrossRefGoogle Scholar
  103. 102.
    Heidemann, G., Bode, M., and Purwins, H. G., Phys. Lett. A 177. 225 (1993).CrossRefGoogle Scholar
  104. 103.
    De Wit, A., Dewel, G., and Borckmans, P., Phys. Rev. E48 R4191 (1993).Google Scholar
  105. 104.
    Gray, P. and Scott, S. K., Chemical Oscillations and Instabilities ( Clarendon Press, Oxford, 1990 ).Google Scholar
  106. 105.
    Doering, C. R. and Horsthemke, W., in Spatio-Temporal Patterns in Nonequilibrium Complex Systems,edited by P. Palffy-Muhoray and P. Cladis (to appear).Google Scholar
  107. 106.
    Ponce Dawson, S., Lawniczak, A., and Kapral, R., J. Chem. Phys. 100, 5211 (1994).CrossRefGoogle Scholar
  108. 107.
    Content, S., Mémoire de Licence, Université Libre de Bruxelles (1993).Google Scholar
  109. 108.
    Content, S., Dewel, G., and Borckmans, P., ‘Absolute and Convective Chemical Instabilities’, preprint (1993).Google Scholar
  110. 109.
    Marek, M. and Svobodova, E., Biophysical Chemistry 3, 263 (1975).CrossRefGoogle Scholar
  111. 110.
    Huerre, P. and Monkevitz, P. H., Ann. Rev. Fluid Mech. 22, 473 (1990).CrossRefGoogle Scholar
  112. 111.
    Deissler, R. J., J. Stat. Phys. 40, 371 (1985).CrossRefGoogle Scholar
  113. 112.
    Rovinsky, A. B. and Menzinger, M., Phys. Rev. Lett. 69, 1193 (1992).CrossRefGoogle Scholar
  114. 113.
    Rovinsky, A. B. and Menzinger, M., Phys. Rev. Lett. 70, 778 (1993).CrossRefGoogle Scholar
  115. 114.
    Spiegel, E. A. and Zaleski, S., Phys. Lett. A 106, 335 (1984).CrossRefGoogle Scholar
  116. 115.
    Doering, C. R. and Horsthemke, W., Phys. Lett. A 182, 227 (1993).CrossRefGoogle Scholar
  117. 116.
    Evans, G. T., J. Theor. Biol. 87, 297 (1980).CrossRefGoogle Scholar
  118. 117.
    Snita, D. and Marek, M., Physica D 75, 521 (1994), and references therein.Google Scholar
  119. 118.
    Kuramoto, Y., Chemical Oscillations, Waves, and Turbulence ( Springer-Verlag, Berlin, 1984 ).CrossRefGoogle Scholar
  120. 119.
    Sivashinsky, G. I., Ann. Rev. Fluid Mech. 15, 179 (1983).CrossRefGoogle Scholar
  121. 120.
    Pelcé, P., Dynamics of Curved Fronts ( Academic Press, San Diego, 1989 ).Google Scholar
  122. 121.
    Horvath, D., Petrov, V., Scott, S. K., and Showalter, K., J. Chem. Phys. 98, 6332 (1993).CrossRefGoogle Scholar
  123. 122.
    Lee, K. J., McCormick, W. D., Ouyang, Q., and Swinney, H. L., Science 261, 192 (1993).CrossRefGoogle Scholar
  124. 123.
    Edblom, E. C., Orban, M., and Epstein, I. R., J. Am. Chem. Soc. 108, 2826 (1986).CrossRefGoogle Scholar
  125. 124.
    Dockery, J. D. and Keener, J. P., SIAM J. Appl. Math. 49, 539 (1989), and references therein.Google Scholar
  126. 125.
    Kness, M., Tuckerman, L. S., and Barkley, D., Phys. Rev. A 46, 5054 (1992).CrossRefGoogle Scholar
  127. 126.
    Kawczynski, A. L., Comstock, W. S., and Field, R. J., Physica D 54, 220 (1992).CrossRefGoogle Scholar
  128. 127.
    Hagberg, A. and Meron, E., Nonlinearity 7, 805 (1994).CrossRefGoogle Scholar
  129. 128.
    Pismen, L. M. and Nepomnyashchy, A. A., Europhys. Lett. 24, 461 (1993).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • P. Borckmans
    • 1
  • G. Dewel
    • 1
  • A. De Wit
    • 1
  • D. Walgraef
    • 1
  1. 1.Service de Chimie-Physique/Center for Nonlinear Phenomena and Complex Systems, CP 231Université Libre de BruxellesBruxellesBelgium

Personalised recommendations